Variation of constants formulae for forward and backward stochastic Volterra integral equations

TL;DR

This paper develops variation of constants formulae for linear forward and backward stochastic Volterra integral equations within Itô's framework, including those with singular kernels and infinitely many iterated integrals.

Contribution

It introduces new notions of products of adapted L^2-processes and resolvent-based formulae, extending the analysis to singular kernels and complex stochastic integral classes.

Findings

Derived variation of constants formulae for linear SVIEs and BSVIEs.

Established duality between generalized SVIEs and BSVIEs.

Included applications to fractional stochastic differential equations.

Abstract

In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual It\^{o}'s framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted $L^2$-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved.